Question

Five cards are drawn successively with replacement from a well-shuffled deck of $52$ cards. Then the probability that all the five cards are spades is:

• A. $\frac{243}{1024}$
• B. $\frac{1}{1024}$
• C. $\frac{1}{512}$
• D. $\frac{81}{512}$

Answer 1

The correct option is B $\frac{1}{1024}$

Let the number of spade cards among the five drawn cards be $X$
As we can observe that the drawing of cards is with replacement, thus, the trials will be bernoulli trials.
Now, we know that in a deck of $52$ cards there are total $13$ spade cards.
Thus, probability of drawing a spade from a deck of $52$ cards
$=\frac{13}{52}=\frac{1}{4}q=1-\frac{1}{4}=\frac{3}{4}$
Thus, $X$ has a binomial distribution with $n=5\text{and}p=\frac{1}{4}$
Thus, $P(X=x){=}^n{C}_x{q}^{n-x}{p}^x,$ where $x=0,1,2,...n$
${=}^5{C}_x{\left(\frac{3}{4}\right)}^{5-x}{\left(\frac{1}{4}\right)}^x$
Probability of drawing all five cards as spades $=P(X=5){=}^5{C}_5{\left(\frac{3}{4}\right)}^0{\left(\frac{1}{4}\right)}^5=1\times \frac{1}{1024}=\frac{1}{1024}$